BackCollege Algebra: Linear Equations, Inequalities, Graphs, and Systems – Study Guide
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1. Linear Equations & Applications
1.1 Solving Linear Equations
Linear equations are algebraic equations of the form ax + b = c, where a, b, and c are constants. Solving these equations involves isolating the variable.
Key Steps: Combine like terms, use inverse operations, and isolate the variable.
Example: Solve
Expand:
Subtract from both sides:
No solution (contradiction).
1.2 Applications of Linear Equations
Word problems can often be modeled with linear equations. Define variables, set up the equation, and solve for the unknown.
Example: A ride-share charges a flat fee of C(h)$.
2. Inequalities
2.1 Solving Linear Inequalities
Linear inequalities are similar to equations but use inequality symbols (<, >, ≤, ≥). The solution is often a range of values.
Key Steps: Solve as you would an equation, but if you multiply or divide by a negative, reverse the inequality sign.
Example: Solve
Solution Representation: Use set-builder and interval notation.
Set-builder:
Interval:
3. Absolute Value Equations
3.1 Solving Absolute Value Equations
The absolute value of a number is its distance from zero on the number line. Equations involving absolute value often have two solutions.
Key Principle: implies or (if ).
Example: Solve
or
or
4. Graphs of Linear Equations
4.1 Slope-Intercept and Point-Slope Form
Linear equations can be written in the form , where m is the slope and b is the y-intercept.
Key Formulas:
Slope:
Point-slope:
Example:
Slope: , y-intercept:
5. Parallel & Perpendicular Lines
5.1 Identifying and Writing Equations
Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals.
Parallel:
Perpendicular:
Example: Find the equation of the line through (4, -1) parallel to .
Slope is (same as given line).
Point-slope:
Simplify to slope-intercept form:
6. Functions
6.1 Evaluating and Combining Functions
A function assigns each input exactly one output. Function notation: . Functions can be combined using addition, subtraction, multiplication, division, or composition.
Example: If , , find .
First,
Then,
7. Systems of Linear Equations
7.1 Solving by Substitution or Elimination
A system of linear equations consists of two or more equations with the same variables. Solutions are points where the equations intersect.
Substitution: Solve one equation for one variable, substitute into the other.
Elimination: Add or subtract equations to eliminate a variable.
Example:
Solution:
8. Applications with Systems
8.1 Word Problems Involving Systems
Many real-world problems can be modeled with systems of equations. Define variables, write equations based on the problem, and solve.
Example: A bookstore sells notebooks for $2 and binders for $6. It sold 120 items for $420. How many of each?
Let = notebooks, = binders
Solution:
Summary Table: Key Properties and Formulas
Concept | Key Formula/Property |
|---|---|
Lines | , , point-slope: |
Parallel | Same slope: |
Perpendicular | Slopes multiply to : |
Absolute Value | or () |
Inequalities | Multiply/divide by negative: flip the sign |
Functions | |
Systems | Substitution or elimination; always verify in both equations |
Additional info: This guide covers foundational topics in College Algebra, including linear equations, inequalities, absolute value, graphing, parallel/perpendicular lines, functions, and systems of equations, with both procedural and application-based examples.
